Let $C$ be a complete curve defined over complex number, $D$ a effective divisor on $C$, and $V\subseteq H^0(D)$ a subspace. Assume that $V$ is base-point-free, which means that the zero of the sections in $V$ do not have common point. Consider the canonical map $V\otimes V \to H^0(2D)$ and denote the image by $W$. It is known that the dimension $\dim\, W \geq 2\dim\,V-1$.
My question is that, can we characterize when the equality holds. I think that if it should be in this way (we assume that $\dim\, V \geq 4$): $\dim\, W = 2\dim\,V-1$ if and only if the image of the map defined by $V$ is a rational curve, i.e., $\varphi_V(C)$ is rational; and $\dim\, W = 2\dim\,V$ if and only if the image of the map defined by $V$ is either a rational curve or an elliptic curve.
